This isn't really a mathematical question, but a question about WA's implementation. Since WA is not open source, we can only guess. I strongly doubt it is using the power series. It is probably just doing the first thing - finding a rational multiple of $\pi$ and doing a lookup.
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Thomas AndrewsMay 16 '14 at 13:01

Can you put your comment into an answer with reasons for your claims?
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user151409May 16 '14 at 13:31

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As described in the documentation: "FunctionExpand uses an extension of Gauss's algorithm to expand trigonometric functions with arguments that are rational multiples of $\pi$."
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BrendonMay 16 '14 at 20:56

3 Answers
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It must be that $\sin\left(\dfrac{p}{q}\pi\right)$ is algebraic. To see why check out this question.

I am almost certain that W|A doesn't use power series unless the value is very small simply because it would be too slow to calculate the value of arbitrary trig functions using power series. It is more likely that there is a certain class of rational numbers such as $\frac{1}{3}$ where the forms of $\sin(\frac{\pi}{3})$ and $\cos(\frac{\pi}{3})$ are known and then formulae such as the double angle formula gives results for other rational numbers such as $\sin({\frac{2\pi}{3}})$. The result is then simplified and sent to the user.

This is only a conjecture as I do not have access to any Mathematica source code.

Wolfram|Alpha possibly just use Mathematica's native capability. About how that's implemented in Mathematica, if you really need to know it, ask it on mathematica.stackexchange.com. My guess is that it uses look-up table for efficiency.